Distributed power allocation for cognitive radio

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DISTRIBUTED POWER ALLOCATION FOR COGNITIVE RADIO Majed Haddad, Merouane Debbah and Aawatif Menouni Hayar

Mobile Communications Group, Institut Eurecom,

2229 Route des Cretes, B.P. 193, 06904 Sophia Antipolis, France E-mail: {haddadm,debbah,menouni}@eurecom.fr

ABSTRACT

In this paper¹, we investigate the idea of usingcognitive radioto reuse locally unused spectrum for communications. We consider a multiband/wideband system with two users in which the primary (licensed) user and the secondary (cognitive) user wish to communicate to the base station, subject to mutual interference. We introduce the notion of the virtual noise-threshold which represents a proxy for the primary user to allow cognitive communications. We determine, under the assumption that each user knows only his own channel, the acceptable interference level within a given quality of service. Moreover, we obtain a characterization of the distributed power allocation for each user, as well as the resulting virtual noise threshold. We prove that a cognitive user can vary its transmit power in order to maximize the sum capacity while maintaining a guarantee of service to the primary user.

1. INTRODUCTION

The recent boom in personal wireless technologies has led to an increasing demand in terms of spectrum resources.

To combat this overcrowding, the Federal Communica- tions Commission (FCC) has been investigating new ways to manage RF resources involving progressive redefini- tion of rules for accessing to the radio spectrum and pos- ing several tasks in the management and in the sharing strategies for such a precious resource. Within this setting, the FCC has recently recommended [1] that significantly greater spectral efficiency could be realized by consider- ingcognitive radio[2]. Such a scheme would define at least two classes of spectrum users. The first would be primary users who already possess a license to use a particu- lar frequency. The second would be secondary (cognitive) users consisting of unlicensed users. Primary users would always have full access to the spectrum when they need it. Secondary users could use the spectrum when it would not interfere with the primary user. Cognitive radio systems offer the opportunity to improve spectrum utilization by detecting unoccupied spectrum bands and adapting the

1Eurecom’s research is partially supported by its industrial partners:

Swisscom, Thales Communications, ST Microelectronics, SFR, Cisco Systems, Sharp, France Telecom, Bouygues Telecom, Hitachi Europe Ltd. and BMW Group Research and Technology. The work reported herein was also partially supported by the French RNRT project GRACE and Idromel.

transmission to those bands while avoiding the interference to primary users. This novel approach to spectrum access is therefore based on reliable detection of primary users and adaptive transmission over a wide bandwidth.

However, there are many challenges across all layers of a cognitive radio system design, from its application to its implementation [3].

In this work, we consider a TDD-uplink communication scenario in which the primary and the secondary user wish to communicate with the base station (BS), subject to mutual interference in a heterogeneous network where devices operate in a wideband/multiband context. One prop- erty of such systems is that, since the same frequency is used, the channel characteristics are nearly the same in both links, provided the channel does not change too rapidly. Under this scheme, we allow the secondary user to transmit simultaneously with the primary user as long as the primary user has not his quality of service (QoS) affected. The motivation of doing so in an environment where two senders share common resources is interference cancelation or mitigation. We derive the power allocations for each of the two users. We show that the overall system capacity can be considerably enhanced by considering cognitive communications.

The rest of the paper is organized as follows: In section 2, we present the channel model system. Section 3 de- scribes the cognitive radio scenario. Section 4 details performance analysis of such systems as well as some simu- lation results. Section 5 concludes the paper.

2. THE CHANNEL MODEL

The baseband discrete-frequency model for uplink channel with two users as described in fig.(1) is:

yⁱ_BS=hⁱ₁ q

P₁ⁱSⁱ₁+hⁱ₂ q

P₂ⁱSⁱ₁+nⁱ, (1) where:

• hⁱ_kis the block fading process of userkfork= 1, 2 on the sub-bandifori=1, ...,N,

• Sⁱ_k is the symbol transmitted by userkon the sub- bandi,

• P_kⁱis the power control of userkon the sub-bandi,

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• nⁱis the additive gaussian noise at theith sub-band.

We statistically model the channelhto be i.i.d distributed over the two fading gains and assume that E

n|h_k|²o

= 1. The additive gaussian noisen at the receiver is i.i.d circularly symmetric andn∼C N^(0,^σ²^).

h2, P2, S² h1, P1, S¹

BS

Primary User Secondary User

. . . . . .

1 N 1 N

Fig. 1. Two-user cognitive radio uplink in a wideband/multiband context.

3. THE COGNITIVE RADIO SCENARIO In current cognitive radio protocol proposals, the device listens to the wireless channel and determines, either in time or frequency, which part of the spectrum is unused.

It then adapts its signal to fill this void in the spectrum domain. Thus, a device transmits over a certain time or frequency band only when no other user does. In this work, the cognitive radio behavior is generalized to allow he secondary user to transmit simultaneously with the primary user as long as the level of interference with the primary user remains within an acceptable range, like in [6]. Specifically, we consider a communication scenario where devices operates in a wideband/multiband context where the two users can be jointly decoded using a suc- cessive interference cancelation (SIC) scheme [4].

On the other hand, in traditional systems (without cognition), when the primary user considers only the ambient noise levelσ², he will maximize his rate byselfishlyex- ploiting all resources through water-filling on this noise level [5]. According to this strategy, primary user leaves no resources for cognitive users to transmit. In our case, the primary user implicity over-estimates the noise level (which he considers as thermal noise plus interference) so as he leaves ”space” for the secondary user. A key idea behind so doing is that, in any case, the primary user will not necessarily need all this rate. Our goal throughout this paper, is to show the feasibility of such approach as well as the capacity gain overall the system with respect to traditional communication systems. Moreover, one basic assumption throughout this contribution is that each user knows only his channel gains. Thus, when channel state information is made available at the primary user, he will

adapt his transmission strategy relative to this knowledge by transmitting at the target rate less than the real data rate with an error-free transmission in order to maintain a guarantee of service. We obtain:

log₂ 1+P₁ⁱ hⁱ₁

2

σ²_v

!

≤log₂ 1+ P₁ⁱ hⁱ₁

2

P₂ⁱ|hⁱ₂|²+σ²

!

Reliable communication can therefore be achieved when the virtual noise threshold is higher than the cognitive in- terferer contributes, yielding:

σ²_v≥P₂ⁱ|hⁱ₂|²+σ², i=1, ...,N (2) Accordingly, the virtual noise threshold has a double role:

(i) it allows cognitive user to profit from primary user resources in an opportunistic manner,

(ii) it maintains a guarantee of service to the primary user when cognitive communication is considered.

We will adopt this framework todistributivelyderive the optimum power allocation policies of each user and ex- amine the variation of the sum capacity as function of the noise-threshold in order to obtain a characterization of the capacity gain by considering cognitive communication. In fact, although cognitive radios have spurred great interest and excitement in industry, many of the fundamental the- oretical questions on the limits of such technology remain unanswered.

4. PERFORMANCE ANALYSIS 4.1. Short Term Analysis

For a given virtual noise-thresholdσ²_v, the maximum short term achievable rate that the primary user can obtain over theNsub-bands is given by:

C1,N= 1 N

N

∑

i=1

log₂

1+P₁ⁱ|hⁱ₁|² σ²_v

(bits/s/Hz) (3) The optimal power allocation which maximizes the transmission rate at the primary user is solution of the optimization problem:

max

P₁¹,...,P₁^N

C1,N, subject to:











1

N∑^N_i=1P₁ⁱ=1,

P₁ⁱ≥0, (4)

In [5] authors looked at the problem of maximizing er- godic capacity subject to an average power constraint, and showed that the optimum power allocation follows from Shannon’s principle of water-filling, namely²:

P₁ⁱ= 1 γ₀− σ²_v

hⁱ₁

2

!+

, i=1, ...,N (5)

2(x)⁺=max(0,x).

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Whereγ0is the Lagrange’s multiplier satisfying the short term average power in (4) with equality.

Let us now focus on the total capacity over the system when a cognitive user is allowed to profit from primary user resources. The expression of the short term sum capacity is given by:

Csum,N= 1 N

N i=1

∑

log₂

1+P₁ⁱ|hⁱ₁|²+P₂ⁱ|hⁱ₂|² σ²

(6) Secondary user offers the opportunity to improve the sum capacity over the system by reliably detecting primary user activity and adapting his transmission while avoiding the interference to the primary user by satisfying constraint in (2) with equality. The motivation of doing so is that if we allow the secondary user to transmit only if he is below a certain interference level, primary user should know the secondary user channel gains to decide at which interference level he should transmit in order to avoid outage. The secondary user power allocation is accordingly solution of the optimization problem:









 1 N

N

∑

i=1

P₂ⁱ=1

P₂ⁱ≥0; i=1, ...,N

σ²_v=P₂ⁱ |hⁱ₂|²+σ²; i=1, ...,N

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The secondary user power allocation that satisfies the con- straints in (7) is therefore:

P₂ⁱ=σ²_v−σ² hⁱ₂

2 ; i=1, ...,N (8) Where the virtual noise threshold is given by:

σ²_v= N

∑^N_i=1 1

|hⁱ₂|²

+σ² (9)

Thus, the secondary user inverses his channel fading in order to guarantee the QoS required to the primary user over theNsub-bands.

4.2. Asymptotic Performances

So far, we have derived power allocations for each user given a virtual noise thresholdσ²_v. Let us now investigate performances of such system when devices operates in a wide-band context, i.e. under the long-term constraint that N→∞. The goal here is to obtain a characterization of the resulting virtual noise threshold and, at the same time, prove the utility of using cognitive radio with respect to traditional communication systems.

By makingNsufficiently large, each user can compute the virtual noise levelindependentlybased only the channel model statistic. In fact, within this setting, the long term average power constraint in (2) becomes:

Z ∞ 0

P₂(t).p(t)dt=1

Where p(t)is the probability density function (p.d.f) of the channel model. By substitutingP2by its expression in (7) we get the following interference constraint onσ²_v:

σ²_v= 1 Z _∞

0

1 t.p(t)dt

+σ² (10)

Notice here that the virtual noise threshold σ²_v depends only the SNR (throughσ²) and the channel statistics (through p(t)). Obviously, such an approach can be immediately translated into results for any non centered p.d.f in order to avoid the non-integrability in zero in (10).

Given this virtual noise level, we will study asymptotical performances of such a system in terms of the sum capacity whenNis assumed to be infinite.

Theorem 1 The sum capacity of cognitive systems using a virtual noise threshold as a proxy for the primary user performs always better than classical communication system (without cognition).

Proof 1 Let us firstly compute the expression of the sum capacity by making N→∞when cognitive communications are possible. The expression in (6) becomes C_sum,∞ =

Z ∞ 0

log₂

1+P₁(t).t+P₂(t).t σ²

.p(t)dt

=log₂ σ²_v

σ² Z _γ₀_σ²_v

0

p(t)dt+ Z _∞

γ0σ²_v

log₂ t

γ₀.σ²_v

.p(t)dt

Whereγ0is the Lagrange’s multiplier satisfying the long- term average power constraint, namely:

1 γ0

Z +∞

γ₀.σ²_v

p(t)dt−σ²_v Z +∞

γ₀.σ²_v

p(t)

t dt=1 (11) Similarly, we compute the sum capacity of a system where the primary user decides to maximize his rate selfishly. In other words, he will water-fill over the ambient noise level σ²and no resources will be left for cognitive users.

C_sum,∞⁰ = Z ∞

0

log₂

1+P1(t).t σ²

.p(t)dt

= Z _∞

γ⁰₀σ²

log₂ t

γ⁰₀.σ²

.p(t)dt.

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Whereγ⁰₀is the Lagrange’s multiplier satisfying the long- term average power constraint on σ². Numerical root finding is needed to determine different values ofγ0and γ⁰₀. Our numerical results, show thatγ0, respectivelyγ⁰₀, increases as σ², respectively σ²_v, decreases³. Now, let us compare the sum capacity in the two configurations.

The difference between the two sum capacities computed above can be written as:

C_sum,∞−C_sum,∞⁰ =log₂ σ²_v

σ² Z γ₀σ²_v

0

p(t)dt+Θ

3Since we haveσ²_v≥σ², we can conclude thatγ0≤γ⁰₀.

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Where:

Θ= Z ∞

γ0σ²_v

log₂ t

γ0.σ²_v

.p(t)dt− Z∞

γ⁰₀σ²

log₂ t

γ⁰₀.σ²

.p(t)dt Therefore, we have just to show thatΘis positive. For the sake of simplicity, we make the assumption that we have γ0.σ²_v≤γ⁰₀.σ². We obtain

Θ = Z _γ⁰

0σ2 γ0σ²_v

log₂ t

γ₀.σ²_v

.p(t)dt+

Z ∞ γ⁰₀σ²

log₂

t γ0.σ²_v

−log₂ t

γ⁰₀.σ²

| {z } .p(t)dt log₂_γ0

0.σ² γ₀.σ²_v

Thus, under our assumptions,Θis always positive and the sum capacity of cognitive system performs always better than for traditional systems.

0 2 4 6 8 10 12 14 16 18 20

1 2 3 4 5 6 7

SNR in dB

Capacity in bits/s/Hz

Cognitive Radio System Non Cognitive System Primary user Capacity

Fig. 2. Simulated capacities of the two configurations:

with cognition and without cognition forN= 10.

Such a result shows the feasibility of allowing secondary users using locally unused spectrum for their transmis- sions with dynamic transmit powers and prove the fundamental constraint on the cognitive radios noise-threshold.

In order to analyze the performance of our approach in terms of achievable rates, we model two rice channel mod- els of mean = 1 to satisfy the constraint in (10). We plot capacitiesCsumandC_sum⁰ as function of the SNR. As men- tioned before in theorem 1, it is clear that the cognitive system performs always better than for system where no cognition exists. Figure 2 also depicts primary user capacity expressed in (3). We remark that primary user capacity achieved in a cognitive system, where device water-fills over the virtual noise levelσ²_v, is always lower than for classical system, where device water-fills over the ambient noise levelσ²especially at high SNRs of interest. Figure always better than for system where no cognition exists.

Figure 3 illustrates the variation of the sum capacity for the two configurations. Notice here that by appropriately choosing the virtual noise thresholdσ²_v, one can increase the sum capacity over the system.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

1 1.5 2 2.5 3 3.5

σ_v²

Sum Capacity in bits/s/Hz

Cognitive system Non cognitive system

Fig. 3. Variation of the sum capacity as function of the virtual noise level forN= 100.

5. CONCLUSION

Using a virtual noise-threshold as a proxy for the primary user, we showed that a cognitive radio can vary its transmit power in order to maximize the sum capacity while maintaining a guarantee of service to the primary user. We characterized the variation of the sum capacity of such a system as function of the virtual noise-thresholdσ²_v. We also showed that one can improve the total capacity overall the system by allowing a single cognitive user to share primary user resources with respect to classical communication systems. As a future work, it is of major interest to generalize the problem to multi-user systems in order to characterize the sum capacity gain of such cognitive net- works.

6. REFERENCES

[1] Federal Communications Commission, Cogni- tive Radio Technologies Proceeding (CRTP), http://www.fcc.gov/oet/cognitiveradio/.

[2] J. Mitola, “Cognitive radio: An integrated agent architecture for software defined radio,” Doctor of Technology, Royal Inst. Technol. (KTH), Stockholm, Sweden, 2000.

[3] S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Communications,” IEEE Journal on Se- lected Area in Communications, vol.23, no 2, pp.

201220, Feb. 2005.

[4] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.

[5] T. M. Cover and J. A. Thomas, Elements of Informa- tion Thoery, Wiley, 1991.

[6] N. Hoven and A. Sahai, “Power Scaling for Cognitive Radio”, WirelessCom, June 2005.